# Polyhedrons. Prism, parallelepiped, pyramid

*Polyhedron. Convex polyhedron. Prism. Right prism. Oblique prism.*

Regular prism. Normal (orthogonal) section of a prism. Parallelepiped.

Right parallelepiped. Right-angled parallelepiped. Cube. Pyramid.

Regular pyramid. Truncated pyramid. Regular truncated pyramid.

Regular prism. Normal (orthogonal) section of a prism. Parallelepiped.

Right parallelepiped. Right-angled parallelepiped. Cube. Pyramid.

Regular pyramid. Truncated pyramid. Regular truncated pyramid.

**
Polyhedron
**
is

*a body, boundary of which consists of pieces of planes (polygons).*These polygons are called

*faces*, their sides –

*edges*, their vertices –

*vertices of*

*polyhedron.*Segments, joining two vertices, which are not placed on the same face, are called

*diagonals of polyhedron.*A polyhedron is called a

*convex*one, if all its diagonals are placed inside of it.

**
Prism
**
is a polyhedron ( Fig.79 ), two faces of which ABCDE and

*abcde*(

*bases*

*of prism*) are equal polygons with correspondingly parallel sides, and the rest of the faces (A

*ab*B, B

*bc*C etc.) are parallelograms, planes of which are parallel to a straight line (A

*a*, or B

*b*, or C

*c*etc.). Parallelograms A

*ab*B, B

*bc*C etc. are called

*lateral faces*; edges A

*a*, B

*b*, C

*c*etc. are

**called**

*lateral edges. A height*

*of prism*is

*any*perpendicular, drawn from any point of one base to a plane of another base. Depending on a form of polygon in a base, the prism can be correspondingly: triangular, quadrangular, pentagonal, hexagonal and so on. If lateral edges of a prism are perpendicular to a base plane, this prism is a

*right prism*; otherwise it is an

*oblique prism*. If a base of a right prism is a regular polygon, this prism is also called a

*regular*one. On Fig.79 an oblique pentagonal prism is shown.

**
Parallelepiped
**
is

*a prism, bases of which are parallelograms.*So, a parallelepiped has six faces and all of them are parallelograms. Opposite faces are two by two equal and parallel. A parallelepiped has four diagonals; they all intersect in the one point and they are divided in it into two. If four lateral faces of parallelepiped are rectangles, it is called a

*right*parallelepiped. A right parallelepiped, all six faces of which are rectangles, is called a

*right-angled*parallelepiped. A diagonal of right-angled parallelepiped

*d*and its edges

*a, b, c*are tied by the relation:

*d*

^{ 2 }

*=*

*a*

^{ 2 }

*+ b*

^{ 2 }

*+ c*

^{ 2 }.

*A right-angled parallelepiped, all faces of which are squares, is called a*

*cube*. All edges of a cube are equal.

**
Pyramid
**
is a polyhedron, one face of which (

*a base of pyramid*) is an arbitrary polygon ( ABCDE, Fig.80 ), and all the rest of the faces (

*lateral*

*faces*) are triangles with a common vertex S, called a

*vertex of a pyramid*. The perpendicular SO, drawn from a vertex of a pyramid to its base, is called a

*height of pyramid*. Depending on a form of polygon in a base, the pyramid can be correspondingly: triangular, quadrangular, pentagonal, hexagonal and so on. A triangular pyramid is a

*tetrahedron,*a quadrangular one – a

*pentahedron*etc. A pyramid is called a

*regular*one, if

*its base is a regular polygon*and its

*height*

*falls into a center of a base.*All lateral edges of a regular pyramid are equal; all lateral faces are equal isosceles triangles. A height of lateral face ( SF ) is called an

*apothem*of a regular pyramid.

If to draw the section
*
abcde
*
, parallel to the base ABCDE ( Fig.81 ) of the pyramid, then a body, concluded between these planes and lateral
surface, is called a
*
truncated pyramid.
*
Parallel faces ABCDE and
*
abcde
*
are called its
*
bases
*
;
a distance O
*
o
*
between them is a
*
height
*
. A truncated pyramid is called a regular one, if a pyramid, from which it was received, is regular.
All lateral faces of a regular truncated pyramid are equal isosceles trapezoids. The height F
*
f
*
of a lateral face ( Fig.81 ) is called
an
*
apothem
*
of a regular truncated pyramid.